A. Direct Modulation vs. External Modulation of Semiconductor Laser Output
The continuing increase of transmission rates at all levels of telecommunication networks and fiber-based RF photonic systems raises demand for very high-speed, low cost optical transmitters. Much effort has been put into developing wide-bandwidth lasers and modulators over the past ten years. To date, the largest reported bandwidth of directly modulated free-running semiconductor lasers at 1.55 μm is 30 GHz, as measured in a Fabry-Perot edge-emitting buried-heterostructure multiple-quantum-well (MQW) laser [Matsui 1997] and in a DFB laser [Kjebon 1997]. On the other hand, external modulators operating at speeds of 40 Gb/s are currently available commercially [Covega 2008] and modulators operating at speeds in the 100-GHz range are under development [Chang 2002]. The widest reported 3-dB modulation bandwidth for Ti:LiNbO3 electro-optic (EO) modulators is 70 GHz, with the maximum measured frequency of 110 GHz [Noguchi 1998]. The drawback of the Ti:LiNbO3 modulators, however, is their poor sensitivity, as represented by their unattractively high half-wave voltage Vπ. Very high modulation frequency and broad-band performance of the Ti:LiNbO3 modulators come at the expense of too high Vπ, which makes them less attractive for system applications [Cox 2006]. A very impressive 145 GHz modulation bandwidth has been demonstrated for a PMMA/DR1 polymer EO modulator at 1310 nm [Lee 2002]. However, the technology of polymer modulators is still very immature, with most of the development effort being focused on the polymer material itself. In general, the polymers with larger EO effect are the least stable against temperature and optical power, which casts doubt on long-term stability of polymer materials [Cox 2006]. In addition, the frequency response of any EO modulator is typically determined by the electrode RF propagation loss and the phase mismatch between the optical beam and modulation microwave [Chung 1991], [Gopalakrishnan 1994], [Chang 2002], which makes overall design and fabrication of these devices complex and costly. Therefore, low-cost small-size directly modulated laser sources with very high modulation bandwidths exceeding 100 GHz are still highly desirable for the rapidly growing applications of RF optical fiber links, and could revolutionize the future of optical telecommunication.
B. Enhancement of Modulation Bandwidth in Injection-Locked Semiconductor Lasers
Since their inception, semiconductor lasers have been key components for many applications in optical fiber communication because of their excellent spectral and beam properties and capability to be directly modulated at very high rates. However, their frequency response has limited the commercial use of directly-modulated lasers to digital transmission not exceeding 10 Gb/s. The modulation response of a diode laser is determined by the rate at which the electrons and holes recombine in the active region (spontaneous carrier lifetime τsp), and the rate at which photons can escape from the laser cavity (photon lifetime τp). The modulation bandwidth is limited by the relaxation oscillation frequency fRO of the laser given by [Lau 1985]2πfRO=√{square root over (gNγpP0)},  (1)where gN is the differential optical gain, P0 is the average photon number in the laser cavity, and γp is the photon decay rate given by the reciprocal of τp. Eq. (1) suggests that the relaxation oscillation frequency can be increased by proper design of laser parameters to get either higher photon density or shorter photon lifetime. Increased injection currents for higher P0 values and shorter laser cavities for smaller τp are ordinarily employed for that purpose in diode lasers. Both approaches, however, involve higher injection current densities, which could result in optical damage to the laser facets and excessive heating. Safe levels of injection current therefore limit the modulation bandwidth in semiconductor lasers. To date, the highest reported experimental relaxation oscillation frequency for a solitary edge-emitting laser is ˜24.5 GHz [Matsui 1997], and ˜15 GHz for a vertical-cavity surface emitting laser (VCSEL) [Lear 1997].
Optical injection locking has been shown to be an extremely effective method to improve microwave performance and linearity of diode lasers and to reach beyond the record values of fRO achieved for free-running devices. Injection locking was first demonstrated in 1976 using edge-emitting lasers [Kobayashi 1976], and in 1996 for VCSELs [Li 1996]. The technique uses output of one laser (master) to optically lock another laser (slave), which can still be directly modulated. Significant increase in the resonance frequency and modulation bandwidth, with reduction in nonlinear distortions [Meng 1999] and frequency chirp [Mohrdiek 1994] has been achieved by injecting external light into diode lasers. So far, improved microwave performance has been observed in edge-emitting lasers with Fabry-Perot cavity [Simpson 1995], [Simpson 1997], [Jin 2006], distributed feedback (DFB) lasers [Meng 1998], [Hwang 2004], [Sung 2004], [Lau 2008b], and VCSELs [Chrostowski 2002], [Chrostowski 2003], [Okajima 2003], [Chang 2003], [Zhao 2004], [Zhao 2006], [Chrostowski 2006a], [Chrostowski 2006b], [Wong 2006], [Lau 2008b]. The highest experimentally observed fRO in excess of 100 GHz was reported for injection-locked DFB lasers and VCSELs, with a record 3-dB bandwidth of 80 GHz being achieved in injection-locked VCSELs [Lau 2008b].
Many aspects of the injection-locking experimental results have been reproduced in analytical studies [Luo 1991], [Simpson 1996], [Nizette 2002], [Nizette 2003], [Lau 2007], [Lau 2008a] and numerical simulations using rate equation models [Luo 1990], [Luo 1992a], [Luo 1992b], [Liu 1997], [Jones 2000], [Chen 2000], [Murakami 2003], [Wieczorek 2006]. Dynamic behavior of diode lasers is described by a system of coupled nonlinear differential equations for the optical field and carrier density in the laser cavity. While for a free-running laser these equations exhibit only damped oscillations with corresponding relaxation oscillation frequency and damping rate, external optical injection increases the number of degrees of freedom by one, which leads to a much greater variety of dynamic behavior. In particular, perturbation analysis of rate equations [Simpson 1996], [Simpson 1997] revealed that the enhanced resonance frequency (the peak frequency in the modulation spectrum) was identical to the difference between the injected light frequency and a shifted cavity resonance, which agreed well with experimental observations. The physical mechanism behind this effect was further clarified in [Murakami 2003], [Wieczorek 2006]. Under strong optical injection, a beating between the injected light frequency and the cavity resonant frequency dominates the dynamic behavior.
FIG. 1 presents a simple illustration of cavity effects and emission frequency in an injection-locked single-mode semiconductor laser. The optical gain spectrum and longitudinal mode spacing are assumed to be sufficiently broad to cover the frequency range of interest. A positive detuning Δωinj=ωinj−ω0 is assumed between the resonant angular frequency ω0 of a solitary (free-running) laser and the angular frequency ωinj of the injected field (FIG. 1a). When the laser is in steady state and locked (FIG. 1b), it emits all its power at the injected frequency ωinj. The cavity resonance, however, must shift to lower frequency by Δω(N), because the refractive index of the active medium increases with the carrier density decrease, and the carrier density N is reduced below its uninjected threshold value due to optical injection. The shift in the carrier-dependent cavity resonance Δω is given by [Lang 1982], [Mogensen 1985]:
                                          Δ            ⁢                                                  ⁢                          ω              ⁡                              (                N                )                                              =                                                    α                2                            ⁢                              v                                  g                  ,                  eff                                            ⁢              Δ              ⁢                                                          ⁢                              G                ⁡                                  (                  N                  )                                                      =                                          α                2                            ⁢                              v                                  g                  ,                  eff                                            ⁢                              G                N                            ⁢              Δ              ⁢                                                          ⁢              N                                      ,                                  ⁢                  (                                    Δ              ⁢                                                          ⁢              N                        =                          N              -                              N                th                                              )                ,                            (        2        )            with α—the linewidth broadening factor, vg,eff—the effective group velocity, Nth—the threshold carrier density, and G—the modal gain, assumed to vary linearly with the carrier density G(N)=GN(N−N0), where GN is the differential modal gain and N0 is the transparency carrier density. In steady state, the gain is too small to support the shifted-frequency mode, and lasing can occur only at the locked frequency ωinj, provided ωinj−ωshift remains within the stable locking range. In the transient process, however, the gain may become sufficient to sustain a mode at the shifted cavity resonance ωshift. Thus, under modulation conditions, the slave laser output may exhibit a damped oscillation at the beat frequency ωinj−ωshift due to interference between those two fields. According to Eq. (2), the resonant frequency produced by this transient interference is given by
                              ω          res                =                                            Δ              ⁢                                                          ⁢                              ω                inj                                      -                          Δ              ⁢                                                          ⁢                              ω                ⁡                                  (                  N                  )                                                              =                                    Δ              ⁢                                                          ⁢                              ω                inj                                      -                                          α                2                            ⁢                              v                                  g                  ,                  eff                                            ⁢                              G                N                            ⁢              Δ              ⁢                                                          ⁢                              N                .                                                                        (        3        )            The most comprehensive study of the modification of dynamical properties of a semiconductor laser by a strong injected signal was reported in [Wieczorek 2006]. Bifurcation theory and continuation techniques were used to explore a wide range of experimental situations involving different injection conditions and types of lasers. The following system of coupled rate equations for injection-locked diode lasers was found to reproduce adequately many aspects of modulation-bandwidth enhancement found experimentally in injection-locked VCSELs:
                                                        ⅆ              E                                      ⅆ              t                                =                                                    1                2                            ⁢              Γ              ⁢                                                          ⁢                              v                                  g                  ,                  eff                                            ⁢                                                g                  N                                ⁡                                  (                                      N                    -                                          N                      th                                                        )                                            ⁢              E                        +                          κ              ⁢                                                          ⁢                              E                inj                            ⁢              cos              ⁢                                                          ⁢              φ                                      ,                            (        4        )                                                                    ⅆ              φ                                      ⅆ              t                                =                                    2              ⁢              π              ⁢                                                          ⁢              Δ                        ⁢                                                  -                                          α                2                            ⁢              Γ              ⁢                                                          ⁢                              v                                  g                  ,                  eff                                            ⁢                                                g                  N                                ⁡                                  (                                      N                    -                                          N                      th                                                        )                                                      -                          κ              ⁢                                                          ⁢                                                E                  inj                                                  E                  ⁢                                                                                                    ⁢              sin              ⁢                                                          ⁢              φ                                      ,                            (        5        )                                                                    ⅆ              N                                      ⅆ              t                                =                                                    j                mod                            ed                        -                                          γ                N                            ⁢              N                        -                                                                                                      ɛ                      0                                        ⁢                                          n                                              g                        ,                        eff                                                              ⁢                    c                                                                              2                      ⁢                                                                                          ⁢                      hv                                        ⁢                                                                                                                ⁡                                  [                                                            g                      th                                        +                                                                  g                                                  N                          ⁢                                                                                                                                                    ⁡                                              (                                                  N                          -                                                      N                            th                                                                          )                                                                              ]                                            ⁢                              E                2                                                    ,                            (        6        )            where E is the intracavity electric field amplitude, φ is the phase difference between the injected and intracavity fields, Γ is the optical confinement factor, κ is the coupling rate coefficient, Einj is the amplitude of the injected field incident upon the slave cavity, Δ is the detuning between injected and free-running laser frequencies, jmod is the modulated pumping current density, e is the electron charge, d is the active region thickness, γN is the electron population decay rate, ε0 is the permittivity of vacuum, ng,eff is the effective group index, h is Planck's constant, ν is the frequency of the intracavity field, and gth is the threshold gain. Injection of a coherent field, with amplitude Einj and frequency νinj, introduces driving terms in the laser field equations, as shown in Eqs. (4), (5) [Spencer 1972]. The threshold gain and carrier density in a free-running laser are
                                          g            th                    =                                                    n                                  g                  ,                  eff                                            ⁢                              γ                p                                                    c              ⁢                                                          ⁢              Γ                                      ;                                  ⁢                              N            th                    =                                    N              0                        +                                                                                n                                          g                      ,                      eff                                                        ⁢                                      γ                    p                                                                    c                  ⁢                                                                          ⁢                  Γ                  ⁢                                                                          ⁢                                      g                    N                                                              .                                                          (        7        )            For sinusoidal modulation, the pumping current density jmod in Eq. (6) can be written asjmod=j0└1+δ sin(2πft)┘,  (8)where j0 is the pre-bias current density, δ is the modulation depth, and f is the modulation frequency. Eqs. (4)-(6) assume single-mode operation, typical for VCSELs or DFB lasers. No theoretical limitation has been found for further increase in the relaxation oscillation frequency fRO. It should be noted, however, that a higher fRO does not necessarily imply a broader modulation bandwidth. A large relaxation oscillation damping rate γRO can undermine the ability of the laser to respond to fast modulation. Hence, a combination of high fRO and low γRO is required for a broad modulation bandwidth. In practice, directly modulated lasers often suffer from high distortions near the resonance frequency fRO, which makes them useful only at RF frequencies much lower than fRO [Lau 1984]. A common practice to increase fRO is to pump the laser high above threshold. However, fRO and γRO are linked in a free-running laser, so any increase in fRO is accompanied by a greater increase in γRO. One very important finding of [Wieczorek 2006] is that coherent optical injection can be used to break that link. For some combinations of injection strength and detuning, fRO can be made to increase, while γRO can remain constant or even slightly decrease.
Another factor limiting modulation bandwidth in injection-locked lasers, evident in all strong-injection-locking experiments, is a sharp roll-off of their modulation response that occurs at low modulation frequencies, before the modulation response gets enhanced by the resonance frequency. The cause of this “sagging” low-frequency response has recently been identified [Lau 2008a] as decoupling of the carrier injection rate from the relaxation oscillation dynamics under strong injection conditions. The cut-off frequency of the low-frequency roll-off can be approximated as [Lau 2008a]:
                                          ω            p                    ≈                                    [                              1                +                                                      ω                    res                                          -                      2                                                        ⁢                  α                  ⁢                                                                          ⁢                                      γ                    p                                    ⁢                  κ                  ⁢                                                                                    S                        master                                                                    S                        0                                                                              ⁢                                      sin                    ⁡                                          (                                              -                                                  φ                          0                                                                    )                                                                                  ]                        ⁢                          g              N                        ⁢                          S              0                                      ,                            (        9        )            where φ0 is the injection-locked phase difference between the injected and intracavity fields, S0 is the photon number in the slave cavity, and Smaster is the number of photons incident from the master laser. The sine term approaches unity as the positive detuning increases. However, the resonance frequency ωres goes up, forcing ωp to smaller values. Two design parameters can be used to maximize ωp: 1) higher differential gain gN, and 2) increased optical power in the slave cavity. We note that a larger α parameter, a higher coupling efficiency (increased coupling rate coefficient κ), and increased power from the master laser Smaster would all result in an increased ωres according to Eqs. (3)-(5), hence they would be ineffective in increasing ωp. The most straightforward method to maximize ωp is to increase internal optical power of the slave laser. By increasing the slave laser bias current from 1.3×Ith to 5×Ith, a very significant improvement in 3-dB bandwidth from ˜1 GHz to a record ˜80 GHz (corresponding to ωres=68 GHz) has been demonstrated in injection-locked VCSELs [Lau 2008b].C. Injection-Locked VCSELs vs Microring Lasers
As described in Section B, injection locking has been actively researched for its potential to improve ultrahigh frequency performance of semiconductor lasers for both digital and analog applications, with VCSELs demonstrating the record high values for enhanced modulation bandwidth. VCSELs were considered to be particularly attractive as injection-locked transmitters because of: 1) short cavity length, leading to a high coupling efficiency, 2) single-mode operation, and 3) low power, resulting in increased injection ratio Pmaster/Pslave (Pmaster is the incident optical power and Pslave is the VCSEL output power) when a master laser with relatively high power is used. The coupling rate coefficient κ, as given by [Schunk 1986] for standard Fabry-Perot lasers, is:κ=c√{square root over (1−R)}/(2ng,effL),  (10)where R is the reflectivity of the laser mirror through which the light is injected and L is the cavity length. Thus, a short cavity length results in a higher coupling efficiency. However, in order to keep the lasing threshold at a reasonably low level, VCSELs require very high mirror reflectivity R, which according to Eq. (10) would bring κ down to a very small value. The value of κ≈1×1012 s−1, estimated using Eq. (10), was reported for VCSELs used in [Chrostowski 2006b].
While the parameter Pmaster/Pslave, defined in [Chrostowski 2006b] as the ratio of the optical power incident on the VCSEL and the output power of the free-running VCSEL, is easy to determine experimentally, it is the ratio of injected power and internal power in the active region of the slave laser that determines its behavior. For example, the stable locking range is given by [Henry 1985]
                                                        -                              c                                  2                  ⁢                                      n                                          g                      ,                      eff                                                        ⁢                  L                                                      ⁢                                                            S                  inj                                S                                      ⁢                                          1                +                                  α                  2                                                              <                      Δ            ⁢                                                  ⁢                          ω              inj                                <                                    c                                                2                  ⁢                                      n                                          g                      ,                      eff                                                        ⁢                  L                                ⁢                                                                                        ⁢                                                            S                  inj                                S                                                    ,                            (        11        )            where Sinj is the number of photons injected from the master laser and S is the number of photons inside the slave laser cavity. Eq. (11) can also be rewritten in terms of the incident power Pmaster and the coupling rate coefficient κ as
                                                        -              κ                        ⁢                                                            P                  master                                P                                      ⁢                                          1                +                                  α                  2                                                              <                      Δ            ⁢                                                  ⁢                          ω              inj                                <                      κ            ⁢                                                            P                  master                                                  P                  ⁢                                                                                                                            ,                            (                  11          ⁢          a                )            where P is the slave laser internal power. Both the wavelength detuning range Δωinj for stable injection locking (>2 nm) and enhanced resonant frequency ωres (up to ˜107 GHz) have been reported to increase with the injection ratio, with no upper limit observed within the instrumentation limit. This is in good agreement with Eq. (3), where both terms on the right-hand side can be made to increase with increased injection rate.
The steady-state analysis of Eqs. (4)-(6) [Murakami 2003] revealed that two parameters are of key importance for achieving highly enhanced modulation bandwidth (for a given internal power of the slave laser)—the cavity roundtrip time τrt=2ng,effL/c and the reflectivity R of the cavity mirror used for injection. Under steady-state injection-locking conditions, the right-hand side of Eq. (3) can be written as
                                          ω            res                    =                                                    -                κ                            ⁢                                                          ⁢                                                E                  inj                                                                      E                    0                                    ⁢                                                                                                    ⁢              sin              ⁢                                                          ⁢                              φ                0                                      =                                          -                                                      c                    ⁢                                                                  1                        -                        R                                                                                                  2                    ⁢                                          n                                              g                        ,                        eff                                                              ⁢                    L                                                              ⁢                                                E                  inj                                                  E                  0                                            ⁢              sin              ⁢                                                          ⁢                              φ                0                                                    ,                            (        12        )            with the phase difference φ0 between the injected and intracavity fields given by
                              φ          0                =                                            sin                              -                1                                      ⁢                          {                                                -                                                            Δ                      ⁢                                                                                          ⁢                                              ω                        inj                                                                                    κ                      ⁢                                                                        1                          +                                                      α                            2                                                                                                                                              ⁢                                                      E                    0                                                        E                    inj                                                              }                                -                                    tan                              -                1                                      ⁢                          α              .                                                          (        13        )            As the frequency detuning moves to the positive edge of the locking range in Eq. (11), ω0 approaches −π/2, thereby increasing ωres [Mogensen 1985], [Murakami 2003]. The smallest possible values for both τrt and R (maximizing the coupling rate coefficient κ) would be ideal for reaching the ultimate limits of modulation bandwidth enhancement in injection-locked lasers. The inherent design trade-off between these parameters, however, makes further optimization of both edge-emitting lasers and VCSELs for enhanced high-speed performance very problematic. While injection-locked VCSELs benefit greatly from very short cavities and, hence, very small τrt, their high-speed performance, at the same time, is compromised by very high mirror reflectivity of a typical VCSEL, resulting in coupling rate coefficients similar to edge emitters. Further improvement of modulation bandwidth in injection-locked VCSELs is expected to come solely from higher power master lasers used for optical injection [Lau 2008b]. For this reason, more complicated cascaded schemes have been attempted, with demonstrated improvement in modulation bandwidth as compared to solitary injection-locked VCSELs [Zhao 2007]. The cascaded optical injection locking is a very promising technique that has scaling-up potential to eventually reach very wide modulation bandwidth over 100 GHz by cascading more slave lasers in a daisy chain structure, as long as the master laser has enough power to stably lock the slave laser with the largest detuning value [Zhao 2007]. This, however, can hardly be realized with VCSELs, notable for their very high mirror reflectivity. In addition, stand-alone VCSELs pose a very serious alignment problem in injection-locking experiments and, at the same time, are not suitable for monolithic integration when injection locking is the requirement. We believe VCSELs are very hard to be optimized for any further improvement in their speed.